So, I'm not sure if I should be going for publication (if so, how and where), or production (again, no centralized place to identify those theaters wanting submissions).
I guess I'm saying that now I'm finished with it, I don't know what to do with it.
Definitely pursue productions. That's the point. Publication can come afterward.
I'm sorry, I don't have a centralized source to identify theaters wanting submissions, although I have a feeling that at least one does exist.
Where are you located?
For major U.S. regional theatres, see
www.lort.org.
Looks like most of these theatres want either scripts submitted through an agent or else a query letter and sample pages and/or synopsis. Check the specifics for the ones you're interested in.
Look for playwriting contests and enter your script in the ones it's appropriate for.
Find smaller semi-professional theatre companies near you, find out whether they have specific policies for script submission, and comply appropriately.
Check with theatre departments at nearby colleges/universities, or for an actor's or writer's center in a major city, for opportunities to get rehearsed staged readings of your work, if not full productions.
Chances are you won't make any money from the first production. But having had one, especially one that gets newspaper reviews, will make it that much easier to get other companies to consider your script.